Point P is in the interior of the equilateral triangle ABC. If AP = 7, BP = 5, and CP = 6, what is the area of ABC?

What is the area of the triangle in the given image, if AP=4, BP=5 and CP=3?

Let ABC be an equilateral triangle. Let P be any point on its incircle. Prove that:

AP2 + BP2 + CP2 = k for some constant k

ABCD is a square and point P inside the square is such that PCD is an equilateral triangle. Find the angle α

P is a point inside triangle ABC. Lines are drawn through P parallel to the sides of the triangle. The three resulting triangles with the vertices at P have areas 4, 9, 49 sq units. The area of triangle ABC is